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Crowd dynamics and conservation laws with non-local constraints and capacity drop
Boris Andreianov, Carlotta Donadello, Massimiliano Rosini
To cite this version:
Boris Andreianov, Carlotta Donadello, Massimiliano Rosini. Crowd dynamics and conservation laws with non-local constraints and capacity drop. Mathematical Models and Methods in Applied Sci- ences, World Scientific Publishing, 2014, 24 (13), pp. 2685-2722. �10.1142/S0218202514500341�. �hal- 00816449v2�
NON–LOCAL CONSTRAINTS
BORIS ANDREIANOV, CARLOTTA DONADELLO, AND MASSIMILIANO D. ROSINI
Abstract. In this paper we model pedestrian flows evacuating a narrow corridor through an exit by a one–dimensional hyperbolic conservation law with a non–local constraint. Existence and stability results for the Cauchy problem with Lipschitz constraint are achieved by a procedure that combines the wave–front tracking al- gorithm with the operator splitting method. The Riemann problem with piecewise constant constraint is discussed in details, stressing the possible lack of uniqueness, self–similarity andL1loc–continuity. One explicit example of application is provided.
Keywords: Crowd dynamics, constrained hyperbolic PDE’s, non–local constraints AMS subject classification: 35L65, 90B20.
1. Introduction
The theory for constrained conservation laws was introduced by Colombo and Goatin in Ref. [8]. Their results are of interest in many real–life applications, such as vehicular traffic, Refs. [11], [20], pedestrian flows, Refs. [4], [13], telecommuni- cations, supply–chains, etc.
For pedestrians, constraints are usually caused by a direct capacity reduction (door or obstacle) and are of fundamental importance in the calculation of evac- uation times. The first macroscopic model for pedestrian evacuations able to reproduce the fall in the efficiency of an exit when a high density of pedestrians clogs it, was the CR model proposed in Ref. [13] and developed in Refs. [9], [10], [16], [28]; see also Ref. [29]. There, the maximal outflow allowed through the exit is assumed to be a piecewise constant function of the density at the exit, and takes two distinct values, one related to the “standard” case, when the density is less than an assigned threshold, and one related to the case with “panic”, when the density is greater than the threshold. As a result, the fall in the efficiency of the exit has a non–realistic behavior since it is instantaneous when the panic reaches the exit. A more realistic model should reproduce a more gradual decay in the efficiency of the exit as the pedestrians accumulate close to it, see Refs. [27], [31].
Moreover, according to the CR model, once the efficiency of the exit falls down, it remains constant until the very last pedestrian is evacuated. On the contrary, in real life the efficiency of the exit gradually increases as the number of the remaining pedestrians to be evacuated becomes smaller and smaller.
To avoid these drawbacks of the CR model, in this paper we generalize the results proved in Ref. [8] and study the Cauchy problem for a one–dimensional
1
hyperbolic conservation law with non–local constraint of the form
∂tρ+∂xf(ρ) = 0 (t, x)∈R+×R (1a)
f(ρ(t,0±))≤p Z
R−
w(x) ρ(t, x) dx
t∈R+ (1b)
ρ(0, x) =ρ0(x) x∈R . (1c)
Above, ρ = ρ(t, x) ∈ [0, R] is the (mean) density at time t ∈ R+ of pedestrians moving along the corridor parameterized by the coordinatex∈R−. Then,R ∈R+ is the maximal density, f : [0, R] → R is the pedestrian flow with pedestrians moving in the direction of increasingx,p :R+→R+ prescribes the maximal flow allowed through an exit placed in x = 0 as a function of the weighted average density of pedestrians in a left neighborhood of the exit, w : R− → R+ is the weight function used for the average density and ρ0 : R → [0, R] is the initial (mean) density. Finally, ρ(t,0−) denotes the left measure theoretic trace along the constraint implicitly defined by
limε↓0
1 ε
Z +∞
0
Z 0
−ε
|ρ(t, x)−ρ(t,0−)| φ(t, x) dx dt= 0
for all φ ∈ C∞c (R2;R). The right measure theoretic trace, ρ(t,0+), is defined analogously.
Observe that ifw is regular enough and limx→−∞w(x) = 0, then the quantity ξ(t) =
Z
R−
w(x) ρ(t, x) dx (2)
is the solution of the following Cauchy problem for an ordinary differential equation ξ(t) =˙
Z
R−
˙
w(x) [f(ρ(t, x))−f(ρ(t,0−))] dx , ξ(0) = Z
R−
w(x) ρ0(x) dx . In real life, when a very high density of pedestrians accumulate near the exit, the outgoing flow can be very small, but remains strictly positive. For this reason, the efficiency of the exit p is assumed to be always strictly positive. We assume that the weight w is an increasing function with compact support because the efficiency of the exit is more affected by the closest high densities, while it does not take into account “far” densities. In summary, we assume that:
(F) f ∈Lip([0, R]; [0,+∞[), f(0) = 0 =f(R) and there exists ¯ρ ∈ ]0, R[ such that f′(ρ) (¯ρ−ρ)>0 for a.e.ρ ∈[0, R].
(W) w ∈ L∞(R−;R+) is an increasing map, kwkL1(R−;R+) = 1 and there exists iw >0 such thatw(x) = 0 for any x≤ −iw.
(P0) p takes values in ]0, f(¯ρ)] and is a non–increasing map.
Observe that f(ρ) < f(¯ρ) for any ρ 6= ¯ρ and ξ(t) ∈ [0, R], see Fig. 1. In the present work we do not take into account the presence of a panic regime since in (F) we assume that the fundamental diagram [ρ 7→ (ρ, f(ρ))] is bell–shaped.
Indeed, the CR model introduces a flux that results from the juxtaposition of two bell–shaped sub–fluxes corresponding to the two regimes quiet–panic and, therefore, does not satisfy the condition (F). The latter assumption (P0) is the
Figure 1. Examples of functions satisfying conditions(F),(W), (P0)and (P2).
minimal requirement for (1) to be meaningful in the sense of distributions, see Definition 1. While existence for the Riemann problem is proved for piecewise constant p, see (P2) in Sec. 4, we strengthen the assumption on p to a Lips- chitz continuity hypothesis when dealing with the Cauchy problem, see(P1)and Theorem 1 in Sec 2.
We give the definition of solution for problem with nonlocal constraint (1) by extending the definition of entropy weak solution for a constrained Cauchy problem of the form
∂tρ+∂xf(ρ) = 0 (t, x)∈R+×R (3a)
f(ρ(t,0±))≤q(t) t∈R+ (3b)
ρ(0, x) =ρ0(x) x∈R . (3c)
Definition 1. Assume conditions (F), (W), (P0). A map ρ ∈ L∞(R+ × R; [0, R])∩C0(R+;L1loc(R; [0, R])) is an entropy weak solution to(1) if there exists q∈L∞(R+; [0, f(¯ρ)]) such that the following conditions hold:
(1) For every test function φ∈C∞c (R2;R+) and for every k∈[0, R]
Z
R+
Z
R
[|ρ−k|∂tφ+ sign(ρ−k) (f(ρ)−f(k))∂xφ] dx dt (4a) + 2
Z
R+
1− q(t) f(¯ρ)
f(k) φ(t,0) dt (4b)
+ Z
R
|ρ0(x)−k|φ(0, x) dx≥0 , (4c)
and
f(ρ(t,0±))≤q(t) for a.e. t∈R+ . (4d) (2) In addition q is linked to ρ by the relation
q(t) =p Z
R−
w(x) ρ(t, x) dx
for a.e. t∈R+ . (5) If q is given a priori, then (4) is the definition of entropy weak solution to problem (3). This item is precisely the Definition 2.1 in Ref. [1], which is a minor
generalization of the original Definition 3.2 introduced in Ref. [8]. We refer to Proposition 2.6 in Ref. [1] for a series of equivalent formulations of conditions (4).
The lines (4a) and (4c) originate from the classical Kruˇzkov Definition 1 in Ref. [22], in the case of the Cauchy problem with no constraints. Lines (4b) and (4d) account for the constraint. Let us stress that both left and right traces atx= 0 of an entropy weak solution exist (see, e.g., Theorem 2.2 in Ref. [1] which is a reformulation of the results of Refs. [26], [33]).
Our main result is well-posedness for the nonlocal problem (1), see Theorem 1.
We show that under the Lipschitz continuity assumption onp there exists a semi- group (St)t>0 on L∞(R; [0, R]) such that ρ(t,·) = St(ρ0) is the unique solution of (1), and depends continuously on tand ρ0 with respect to theL1loc–distance.
The uniqueness result for (1) is a consequence of a stability estimate for the problem with local constraint (3) with respect to the L1loc–distance, of the rela- tion (5) and of the Gronwall inequality.
The existence result for (1) is achieved through an operator splitting method, Refs. [6], [7], [14], [15], [18], coupled with the wave–front tracking algorithm, Refs. [8], [17]. This procedure is chosen for two reasons. First, wave–front track- ing schemes are able to operate also in the case with panic, when nonclassical shocks away from the constraint have to be taken into account, see Ref. [4]. Sec- ond, the operator splitting procedure allows us to approximate our problem with a problem of type (3), namely with a “frozen” constraint. This greatly simplify our work because it avoids the difficulties coming from the Riemann solver for the nonlocally constrained problem. Indeed, we underline the fact that, differently from the constrained Cauchy problems studied in Refs. [1], [8], the maximal flow at the constraint for (1) depends on the solution itself and, in general, it is an un- known variable of the problem. As soon asp is discretized,i.e.p is approximated by piecewise constant functions, the solution of the corresponding nonlocally con- strained Riemann problem may fail to be unique,L1loc–continuous, consistent and self–similar, as we will show in Sec. 4. Furthermore, using wave–front tracking for a nonlocal problem is quite delicate because one cannot merely juxtapose local solutions of Riemann problems, see Remark 1.
The use of wave–front tracking approximation requires theBV functional set- ting. As already observed in Ref. [8], the constraint may cause sharp increases in the total variation TV(ρ) of the solution. To overcome this difficulty, as in Refs. [5], [8], [11], [32], we rather estimate the total variation of Ψ◦ρ, where
Ψ(ρ) = sign(ρ−ρ) (f¯ (¯ρ)−f(ρ)) = Z ρ
¯ ρ
f˙(r)
dr . (6) We stress that Ψ is one–to–one, but possibly singular atρ = ¯ρ. Indeed, if ρ is in BV, then also Ψ◦ρ is in BV, while the reverse implication does not hold true.
Therefore we introduce the set D=
ρ∈L1(R; [0, R]) : Ψ(ρ)∈BV(R;R) . (7) In the first step of our construction, we exploit the finite speed of propagation property for the conservation law (1a) to define St on the domain D by coupling
the operator splitting method with the wave–front tracking algorithm, see Sec 3.
Then we extendStto L∞ by a density argument, see the second part of the proof of Theorem 1.
The paper is organized as follows. Sec. 2 and 3 are devoted to the constrained Cauchy problem. In Sec. 4 we study (1) with a Riemann initial datum. In Sec. 5 we apply the model (1) to describe the evacuation of a corridor through an exit placed at x= 0. All the technical proofs are in Sec. 6. Conclusions and perspectives are outlined in Sec. 7.
2. The Cauchy problem with nonlocal constraint
In this section we consider the Cauchy problem (1) under the hypotheses (F), (W)and the following assumption on p:
(P1) p belongs to Lip([0, R] ; ]0, f(¯ρ)]) and it is a non-increasing map.
Let us start with the basic properties of entropy weak solutions to (1).
Proposition 1. Let [t7→ρ(t)]be an entropy weak solution of (1)in the sense of Definition 1. Then
(1) It is also a weak solution of (1a), (1c).
(2) Any discontinuity satisfies the Rankine–Hugoniot jump condition.
(3) Any discontinuity away from the constraint is classical, i.e. satisfies the Lax entropy inequalities.
(4) Nonclassical discontinuities, see Refs. [24], [29], may occur only at the con- straint location x = 0, and in this case the flow at x = 0 is the maximal flow allowed by the constraint. Namely, if the solution contains a nonclassical discontinuity for all times t∈I, I open in R+, then for a.e. tin I
f(ρ(t,0−)) =f(ρ(t,0+)) =p Z
R−
w(x) ρ(t, x) dx
. (8)
Proof. By taking k = 0, then k = R, in (4), we deduce that any entropy weak solution to (1) is also a weak solution to (1a), (1c). As a consequence,ρsatisfies the Rankine–Hugoniot jump condition and, in particular, f(ρ(t,0−)) = f(ρ(t,0+)).
By taking in (4) a test function with support inR+×R−, then inR+×R+, we see that ρ is also a classical Kruˇzkov solution to (1a), (1c) in R+×R± and therefore the jumps in ρ located at x 6= 0 satisfy the Lax entropy inequalities. Finally, we prove property (8). This property was observed at the level of problem (3), see in particular the description of the “germ” GF in Ref. [1], but it was not explicitly stated in the works Refs. [1], [8] devoted to problem (3). For the sake of completeness, we give an explicit proof of property (8) for problem (4). Consider the test function
φ(t, x) =
"
Z +∞
|x|−ε
δε(z) dz
# ψ(t) ,
whereψ∈C∞c (R;R+) is such that ψ(0) = 0, while δε is a smooth approximation of the Dirac mass centered at 0+,δD0+, namely
δε∈C∞c (R;R+), ε∈R+, supp(δε)⊆[0, ε], kδεkL1(R;R) = 1, δε→δ0+D . (9) Observe that asεgoes to zero
φ(0, x)≡0→0 , ∂tφ(t, x) =
"
Z +∞
|x|−ε
δε(z) dz
#
ψ(t)˙ →0 , φ(t,0) =ψ(t)→ψ(t) , χR±(x) ∂xφ(t, x)→ ∓δ0±D (|x|) ψ(t) . Then, if we take k= ¯ρ andφas test function in (4), we obtain as εgoes to zero
Z
R+
[Ψ (ρ(t,0+))−Ψ (ρ(t,0−))]ψ(t) dt+ 2 Z
R+
[f(¯ρ)−p(ξ(t))] ψ(t) dt≥0 , whereξ is defined by (2). For the arbitrariness ofψ, we have for a.e. t >0
Ψ (ρ(t,0+))−Ψ (ρ(t,0−)) + 2 [f(¯ρ)−p(ξ(t))]≥0 .
Therefore, if fort∈I the solution has a nonclassical discontinuity at the constraint location x = 0, then by the assumption (F) and the Rankine–Hugoniot jump condition,ρ(t,0+)<ρ < ρ(t,¯ 0−) andp(ξ(t))≤f(ρ(t,0±)) for a.e.t∈I. Finally, by the condition (4d) of Definition 1, it has to bep(ξ(t)) =f(ρ(t,0±)) for a.e.t∈
I.
The following theorem on existence, uniqueness and stability of entropy weak solutions of the constrained Cauchy problem (1) is the main result of this paper.
Theorem 1. Let (F),(W), (P1) hold. Then
(i) For any initial datum ρ0 ∈L∞(R; [0, R]), the Cauchy problem (1) admits a unique entropy weak solution ρ in the sense of Definition 1. Moreover, if
˜
ρ = ˜ρ(t, x) is the entropy weak solution corresponding to the initial datum
˜
ρ0 ∈L∞(R; [0, R]), then for all T >0 and L >iw there holds
kρ(T)−ρ(T˜ )kL1([−L,L];R)≤eCTkρ0−ρ˜0kL1([−(L+M T),(L+M T)];R), (10) where M = Lip(f) and C = 2Lip(p)kwkL∞(R−;R).
(ii) If ρ0 belongs toD, defined as in (7), then the unique entropy weak solution of problem (1) verifies ρ(t,·)∈ D for a.e. t >0, and it satisfies
TV (Ψ (ρ(t)))≤Ct= TV (Ψ (ρ0)) + 4f(¯ρ) +C t , (11) moreover, for a.e. t, s in ]0, T[we have
kΨ (ρ(t,·))−Ψ (ρ(s,·))kL1(R;R) ≤ |t−s| Lip(Ψ) CT . (12) Proof. The proof consists of three parts, the longest one being postponed to Sec. 3.
Uniqueness and stability
Conditions (4) of Definition 1 ensure that for allt∈[0, T] we can apply the stability estimate in Proposition 2.10 in Ref. [1]. More specifically, if ρ and ˜ρ are solutions
of (1) corresponding to the constraintsq and ˜q, and the initial conditions ρ0 and
˜
ρ0, respectively, then
kρ(t)−ρ(t)k˜ L1([−L,L];R)≤ kρ0−ρ˜0kL1({|x|≤L+M t};R)+ 2 Z t
0
|q(s)−q(s)|˜ ds.
By the explicit expression of the constraints q and ˜q, see (5), we have Z t
0
|q(s)−q(s)|˜ ds= Z t
0
p
Z
R−
w(x) ρ(s, x) dx
−p Z
R−
w(x) ˜ρ(s, x) dx
ds, and this quantity is bounded by Lip(p) w(0−)Rt
0kρ(s)−ρ(s)k˜ L1([−iw,0];R)ds by the Lipschitz continuity ofpand H¨older inequality. Note the inclusions [−L, L]⊇ [−iw,0] and{|x| ≤L+M T} ⊇ {|x| ≤L+M t}. We complete the proof by applying Gronwall’s inequality, see for instance Ref. [29].
Existence in D
The existence problem in theD–framework will be addressed in Sec. 3, see Propo- sition 2. With this result in hand, existence inL∞follows by the density argument we develop below.
Existence in L∞
Let ρ0 be in L∞(R; [0, R]). By the standard diagonal procedure argument, it is enough to prove existence on an arbitrary time interval [0, T] in R+. Introduce a sequence Ψn0 in BVloc(R;R) which converges pointwise a.e. to Ψ0 = Ψ(ρ0). Set ρn0 = Ψ−1(Ψn0). For any L ∈ N sufficiently large, set ρn,L0 = ρn0 χ{|x|≤L+M T}. We have that ρn,L0 belongs to D and ρn,L0 converges to ρ0 in L1loc(R; [0, R]) as L and n go to infinity. Let ρn,L be the corresponding entropy weak solution of (1) constructed in Proposition 2. Then, for any L >1 + iw
(A) for all t ∈ [0, T] we have that
ρn,L(t)−ρm,L(t)
L1([−L,L];R) goes to zero as m and ngo to infinity;
(B) if L′ > L, thenρn,L≡ρn,L′ on [0, T]×[−L, L].
Properties (A) and (B) follow by (10). Then, by taking L(x) = ⌊|x|⌋+ 1, prop- erty (A) ensures that we can introduce the functionρ(t, x) = limn→+∞ρn,L(x)(t, x).
By (B) we also have thatρ(t, x) = limn→+∞ρn,L′(t, x) for any L′ > L(x).
We prove now that [t 7→ ρ(t)] is an entropy weak solution to (1) with initial datum ρ0. For any compact set K ⊂ R, take L such that [−L+ 1, L−1] ⊇ (K ∪ [−iw,0]). Then ρn,L converges to ρ in L1([0, T] ×K; [0, R]) and conse- quently for a.e. t ∈ [0, T], qn,L(t) = p
R0
−iww(x) ρn,L(t, x) dx
converges to q(t) =p
R0
−iww(x) ρ(t, x) dx
inL1([0, T];R). This is enough to ensure that for all test functionsφ supported in [0, T]×K, the function ρ satisfies (4a)–(4c) and that (5) holds. In particular the Rankine–Hugoniot condition is satisfied, therefore using Lemma 3, we have that fn ρn,L(·,0−)
converges weakly to f(ρ(·,0−)) in L1([0, T];R) andf(ρ(·,0−)) =f(ρ(·,0+)). Therefore, also (4d) holds true.
3. Wave–front tracking and operator splitting methods
In this section we construct solutions for initial data inD and we prove thatD is an invariant domain for the semigroup S.
Proposition 2. For any initial datumρ0 in D, there exists a unique entropy weak solution of problem (1), [t7→ρ(t)] and ρ(t) belongs to D for all t >0. Moreover, estimates (11) and (12) are satisfied.
The solution [t7→ρ(t)] is the limit (along a subsequence) of a sequence obtained by combining the wave–front tracking algorithm and the operator splitting method.
In the following subsections we describe the construction in full details.
3.1. Approximation of flux and efficiency functions. Fix h, n ∈ N suffi- ciently large with n≫h. Introduce the mesh
Mn=f−1 2−nf(¯ρ)N∩[0, f(¯ρ)]
and the set
Dn=D ∩PC(R;Mn) ,
wherePC(R;Mn) is the set of piecewise constant functions defined onR, taking values inMn and with a finite number of jumps. Approximate the fluxf with a piecewise linear, continuous flux fn : [0, R]→[0, f(¯ρ)], whose derivative exists in [0, R]\ Mn and such thatfncoincides withf onMn, see Fig. 2, left. Clearly,fn satisfies condition (F). Considerp−1 f(Mh)∩p([0, R])
={ξ˜0h, . . . ,ξ˜mhh+2}, with 0≤ξ˜0h<ξ˜1h< . . . <ξ˜hmh+2 ≤R, and observe that
ξ˜i+1h −ξ˜ih
Lip(p)≥p ξ˜hi+1
−p ξ˜hi
= 2−hf(¯ρ) . (13) Approximate p with the functionph∈PC [0, R];f(Mh)
defined as follows:
Figure 2. Left: : In thin linefand in thick line the approximation fn. Right: In thin linep and in thick line the approximation ph.
ph(ξ) =
mh−1
X
i=0
phi χ
ξhi, ξi+1h (ξ) +phmh χ
ξhmh, R(ξ), (14a)
where
0 =ξ0h < ξ1h= ˜ξ1h< . . . < ξhmh= ˜ξmhh < ξmhh+1=R , (14b) phi =p(ξi+1h ), i= 0, . . . , mh−1, andphmh=p(˜ξhmh+1) , (14c) see Fig. 2, right. Since h < n, we have that f(Mh) ⊂ f(Mn), ph([0, R]) ⊆ f(Mn)\ {0},
p−ph L∞
([0,R];R)≤21−hf(¯ρ) (15) and by (13) and (14)
phi −phi+1= 2−hf(¯ρ) , (16)
i=0,...,minf h
ξhi+1−ξih
≥ 2−hf(¯ρ)
Lip(p) . (17)
3.2. The algorithm. Now we can start with the construction of an approximating solution [t 7→ ρn,h(t)] to (1). As a first step we associate to any fractional time interval of the form [ℓ∆th,(ℓ+ 1)∆th[, ∆th > 0, ℓ ∈ N, a constrained Cauchy problem of the form (3) with constant constraint. Then the wave–front tracking algorithm gives us the corresponding exact solution [t7→ ρn,hℓ+1(t)]. Finally, ρn,h is obtained by gluing together ρn,hℓ+1,ℓ∈N. The existence of a limit forρn,hasnand h go to infinity is ensured by the choice
∆th= 1
2h+1w(0−)Lip(p) , (18)
which will be motivated in the proof of Lemma 1 by a sort of CFL condition.
Roughly speaking, this condition is needed to bound the possible jump in the value of the constraint due to the update at each fractional time (ℓ+ 1)∆th,ℓ∈N. Approximate ρ0 with a piecewise constant function ρn0 :R → [0, R] that coin- cides with ρ0 on Mn and such thatkρn0kL1(R;R) ≤ kρ0kL1(R;R) and TV (Ψ (ρn0))≤ TV (Ψ (ρ0)). Clearly, ρn0 belongs to Dn. First consider the approximating con- strained Cauchy problem
∂tρ+∂xfn(ρ) = 0 (t, x)∈]0,∆th[×R fn(ρ(t,0±))≤ph(Ξn0) t∈]0,∆th]
ρ(0, x) =ρn0(x) x∈R,
where
Ξn0 = Z
R−
w(x) ρn0(x) dx .
The unique exact solution [t7→ρn,h1 (t)] for the above problem is obtained by piecing together the solutions to the Riemann problems at points whereρn0 is discontinuous or where interactions take place, namely where two or more waves intersect, or one or more waves reach x = 0. For the definition of solution of the Riemann problem with a piecewise linear, continuous flux away from the constraint, we
refer to Sec. 6.1 in Ref. [2] or to Sec. 5.2 in Ref. [29]. The definition of solution to the constrained Riemann problem along x= 0 follows by the obvious adaptation of Definition 2.2 in Ref. [8] to the case with a piecewise linear continuous flux, see also Sec. 6.3 in Ref. [29]. The results of Theorem 3.4 in Ref. [8] can be easily generalized to the case with piecewise linear continuous flux and, therefore, we can define
ρn,h(t, x) =ρn,h1 (t, x) for (t, x)∈]0,∆th]×R.
We can assume that no interaction occurs at time t = ∆th, see assumption H2 below. Then the approximate solution is prolonged beyond t= ∆th by taking
ρn,h(t, x) =ρn,h2 (t−∆th, x) for (t, x)∈]∆th,2∆th]×R, where [t7→ρn,h2 (t)] is the exact solution of the constrained Cauchy problem
∂tρ+∂xfn(ρ) = 0 (t, x)∈]0,∆th[×R fn(ρ(t,0±))≤ph
Ξn,h1
t∈]0,∆th] ρ(0, x) =ρn,h1 (∆th, x) x∈R, with
Ξn,h1 = Z
R−
w(x) ρn,h1 (∆th, x) dx .
We repeat this procedure at each fractional step and, once we get [t7→ρn,hℓ (t)], we construct [t7→ρn,hℓ+1(t)] by solving a constrained Cauchy problem of the form
∂tρ+∂xfn(ρ) = 0 (t, x)∈]0,∆th[×R (19a) fn(ρ(t,0±))≤ph
Ξn,hℓ
t∈]0,∆th] (19b)
ρ(0, x) =ρn,hℓ (∆th, x) x∈R, (19c)
where
Ξn,hℓ = Z
R−
w(x) ρn,hℓ (∆th, x) dx . (19d) We stress that the solution to (19) is unique and that the efficiency at the exit may change at each time t∈∆thNand only there.
To simplify the wave–front tracking algorithm, see Remark 7.1 in Ref. [2], it is standard to remark that, without loss of generality, one can assume that:
H1 At any interaction either exactly two waves interact, or a single wave reaches the constraint x= 0.
H2 No interaction occurs at time t∈∆thN. In this way we construct
Ξn,h(t) =X
ℓ∈N
Ξn,hℓ χ[ℓ∆th,(ℓ+ 1)∆th[(t) (20)
and an approximate solution of the Cauchy problem (1) ρn,h(t, x) =X
ℓ∈N
ρn,hℓ+1(t−ℓ∆th, x) χ]ℓ∆th,(ℓ+ 1)∆th](t) , (21) where [t7→ρn,hℓ+1(t)] is the unique solution to (19).
Roughly speaking, the present procedure consists in the application of two op- erators, Θ and S, at each fractional step ]ℓ∆th,(ℓ+ 1)∆th], ℓ ∈ N. The first operator gives Ξn,hℓ = Θ[ρn,hℓ (∆th)], while the second operator gives the solution ρn,hℓ+1 = S[ρn,hℓ (∆th),Ξn,hℓ ] of the constrained Cauchy problem of the form (19), with [x7→ρn,hℓ (∆th, x)] as initial datum and with ph(Ξn,hℓ ) as constraint.
More rigorously, for anyρn0 ∈ Dnand t∈R+, define recursively Fn,h[ρn0](t) =S[ρn0,Θ [ρn0]] (t)
if t∈[0,∆th], and, if t∈](ℓ+ 1)∆th,(ℓ+ 2)∆th],ℓ∈N, then Fn,h[ρn0](t) =Sh
Fn,h[ρn0] ((ℓ+ 1)∆th),Θh
Fn,h[ρn0] ((ℓ+ 1)∆th)ii (t) . 3.3. A priori estimates. In this section we prove thatρn,h(t) =Fn,h[ρn0](t) is in Dn on any bounded time interval [0, T],T >0, and we estimate TV Ψ ρn,h(t) uniformly inn,handt. To this aim, we introduce the following Temple functional
Υn,hT (t) = TV Ψ
ρn,h(t) +γh
ρn,h(t),Ξn,h(t)
+ ΓhT (t) , (22) with
γh(ρ,Ξ) =
0 if ρ(0−)>ρ > ρ(0+) and¯ fn(ρ(0±)) =ph(Ξ(t)) 4
f(¯ρ)−ph(Ξ)
otherwise, ΓhT(t) = 5·2−h f(¯ρ)
T
∆th − t
∆th
,
where ⌊·⌋ :R→ Zdenotes the floor function. Recall that the Temple functional adopted in Ref. [8] involves the total variation of the approximating constraint, which is given a priori. In our construction, at each fractional time interval we are dealing with a different approximating problem (19) and we need to know the solution at the previous step in order to fix the value of the constraint in (19b).
Therefore, the constraint p Ξn,h(t)
,t∈R+, and its total variation are not given a priori. Nevertheless, due to the choice of ∆th, we are able to bound the possible jump of p Ξn,h(t)
at each time step, as we will see in Lemma 1, and estimate a priori the total variation of the efficiency. From this point of view, the functional Υn,hT is the natural generalization of that one used in Ref. [8]. In fact, the two functionals have in common the first two terms, namely
Qn,h(t) = TV Ψ
ρn,h(t) +γh
ρn,h(t),Ξn,h(t)
, (23)
while ΓhT (t) is introduced to control the total variation of p Ξn,h(·)
in the time interval [t, T].
Lemma 1. For any ℓ ∈ N, the jump in the efficiency at time t = (ℓ+ 1)∆th, namely
ph(Ξn,hℓ+1)−ph(Ξn,hℓ )
, is either zero or2−hf(¯ρ).
Proof. Fix ℓ ∈ N. If
Ξn,hℓ+1−Ξn,hℓ
< inf
i=0,...,mh
ξi+1h −ξhi
, then [ξ 7→ ph(ξ)] has at most one jump for ξ between Ξn,hℓ+1 and Ξn,hℓ and (16) allows us to conclude.
Because of (17), we just need to show
Ξn,hℓ+1−Ξn,hℓ
Lip(p)<2−hf(¯ρ) .
By Proposition 1,ρn,hℓ+1is a weak solution of the problem (19a), (19c) withρn,hℓ (∆th) as initial condition. Then, for any φinC1c(R2;R) we have
Z
R+
Z
R
h
ρn,hℓ+1 ∂tφ+f(ρn,hℓ+1) ∂xφi
dxdt+ Z
R
ρn,hℓ (∆th, x) φ(0, x) dx= 0. (24) Let (ην)ν be a standard family of mollifiers and define wν =w∗ην. Let δε be as in (9). Take 0≤t1 < t2 ≤∆th and consider the test function
φ(t, x) =
Z t−t1
t−t2+ε
δε(z) dz
Z x+iw
x+ε
δε(z) dz
wν(x) . Observe thatφ(0,·)≡0 and that lettingεgo to zero we get
∂tφ(t, x)→[δtD1(t)−δtD2(t)]χ[−iw,0](x) wν(x) ,
∂xφ(t, x)→χ[t1, t2](t) [δD−iw+(x)−δ0−D (x)]wν(x). We pass to the limit in the Eq. (24) lettingε go to zero and we obtain
Z 0
−iw
wν(x)h
ρn,hℓ+1(t1, x)−ρn,hℓ+1(t2, x)i dx
= Z t2
t1
h
wν(0−) f
ρn,hℓ+1(t,0−)
−wν(−iw+) f
ρn,hℓ+1(t,−iw+)i dt . Then, asν goes to infinity we get
Z 0
−iw
w(x)h
ρn,hℓ+1(t1, x)−ρn,hℓ+1(t2, x)i dx
≤(t2−t1)f(¯ρ)w(0−) . (25) By (19c) and (19d) we have that
Ξn,hℓ+1−Ξn,hℓ =
Z
R−
w(x)h
ρn,hℓ+1(∆th, x)−ρn,hℓ (∆th, x)i dx
= Z
R−
w(x)h
ρn,hℓ+1(∆th, x)−ρn,hℓ+1(0, x)i dx
≤∆th f(¯ρ) w(0−) .
Therefore by (18) the proof is complete.
We are ready to show that Υn,hT is a Temple functional.
Proposition 3. Let h, n ∈N and ρn0 ∈ Dn. On [0, T], the map [t7→ Υn,hT (t)] is non–increasing and it decreases by at least 2−nf(¯ρ) each time the number of waves increases.
The proof is deferred to Sec. 6.1.
In the next corollary we rely on Proposition 3 to prove a uniform estimate on TV Ψ ρn,h(t)
.
Corollary 1. There exists a constantC >0, that does not depend onnor h, such that for all t >0
TV
Ψ
ρn,h(t)
≤TV (Ψ (ρ0)) + 4f(¯ρ) +C t . (26) Proof. We consider the functionalQn,h= Υn,hT −ΓhT introduced in (23). Proceeding as in the proof of Proposition 3, we can show that Qn,h may increase only at t∈∆thN. However, since Υn,hT is strictly decreasing at t∈∆thN, we have that for all ℓ∈N,
Qn,h(ℓ∆th+)− Qn,h(ℓ∆th−)≤
ΓhT(ℓ∆th+)−ΓhT(ℓ∆th−)
= 5·2−hf(¯ρ) . Therefore, by (18)
TV
Ψ
ρn,h(t)
≤ Qn,h(t)≤ Qn,h(0) + 5·2−hf(¯ρ) t
∆th
≤TV (Ψ (ρ0)) + 4f(¯ρ) + 10w(0−) Lip(p) f(¯ρ)t , and the estimate (26) holds withC = 10w(0−) Lip(p) f(¯ρ).
By the results proved in Ref. [8], the assumptionH2 and the corollary above, we have that both ρn,h(t) and Ξn,h(t) are well defined for any t∈[0, T] and that ρn,h belongs to C0(R+;Dn). In particular, [t 7→ ρn,h(t)] is piecewise constant with discontinuities along finitely many polygonal lines with bounded speed of propagation, that do not intersect each other at any time t ∈ ∆thN. By the construction of ρn,h and its continuity with respect to time it is not difficult to show
Proposition 4. The map [t7→ρn,h(t)] given by (21) is an entropy weak solution in the sense of Definition 1 (with fn, ph replacing f, p) to the problem
∂tρ+∂xfn(ρ) = 0 (t, x)∈R+×R (27a) fn(ρ(t,0±))≤ph
Ξn,h(t)
t∈R+ (27b)
ρ(0, x) =ρn0(x) x∈R, (27c)
where [t7→Ξn,h(t)]is given by (20).
The proof is deferred to Sec. 6.2.
Proposition 5. There exists a subsequence ofρn,h converging a.e. on R+×R to a limit ρ∈L∞(R+×R; [0, R]). In addition, ρ satisfies estimates (11) and (12).
Proof. By the standard diagonal procedure argument, it is enough to prove con- vergence on an arbitrary time interval [0, T], T > 0. The sequence Ψ ρn,h
is uniformly bounded in L∞([0, T]×R;R)∩L∞([0, T];BV(R;R)) by Corollary 1.
In order to get compactness in L1loc, see Theorem 2.4 in Ref. [2], we still need to show that [t7→Ψ ρn,h(t,·)
] is Lipschitz with respect to theL1–norm. In analogy to (6), we define
Ψn(ρ) = sign(ρ−ρ) (f¯ n(¯ρ)−fn(ρ)) = Z ρ
¯ ρ
f˙n(r)
dr . (28) We observe that Ψ coincides with Ψn on Mn and, as a consequence, Lip(Ψn) ≤ Lip(Ψ). Since ρn,h takes values in Mn, we have Ψn ρn,h
= Ψ ρn,h
. Hence, TV Ψn ρn,h(t)
= TV Ψ ρn,h(t)
and by Corollary 1 we have
∂xΨn ρn,h
L∞
([0,T];Mb(R;R))≤CT = TV (Ψ (ρ0)) + 4f(¯ρ) +C T , uniformly in n and h. Above, Mb(R;R) denotes the space of bounded Radon measures. Let gn=fn◦(Ψn)−1 and remark that by (28)
˙
gn(ψ) = f˙n◦(Ψn)−1(ψ)
Ψ˙n◦(Ψn)−1(ψ) = f˙n◦(Ψn)−1(ψ)
f˙n◦(Ψn)−1(ψ)
∈ {−1,1} .
Hence ∂tρn,h is bounded inL∞([0, T];Mb(R;R)) because, by Eq. (27) and Theo- rem 4 in Ref. [21], we have
∂tρn,h
L∞([0,T];Mb(R;R)) ≤ kg˙nkL∞([−f(¯ρ),f(¯ρ)];R)
∂xΨn ρn,h
L∞([0,T];Mb(R;R))
≤CT .
As the functions Ψn are uniformly Lipschitz, also the distributions µn,h =
∂tΨn(ρn,h) are uniformly bounded measures in L∞([0, T];Mb(R;R)) with
µn,h
L∞([0,T];Mb(R;R))≤Lip(Ψn) CT .
Now, let (ην)ν be a standard family of mollifiers inC∞c (R2;R) and defineFνn,h= Ψn ρn,h
∗ην andµn,hν =µn,h∗ην. Then
µn,hν
L∞
([0,T];L1(R;R))≤ µn,h
L∞
([0,T];Mb(R;R)) .
Due to the regularity of Fνn,h, for any δ >0 and for any 0≤t < t+δ≤T
Fνn,h(t+δ,·)−Fνn,h(t,·) L1
(R;R)= Z
R
Z t+δ
t
µn,hν (s, x) ds
dx
≤δ µn,hν
L∞([0,T];L1(R;R)) ≤δ Lip(Ψn) CT ,
and asνgo to zero we deduce the uniform Lipschitz continuity in time of Ψ ρn,h
= Ψn ρn,h
:
Ψ(ρn,h(t+δ,·))−Ψ(ρn,h(t,·)) L1
(R;R)≤δ Lip(Ψ) CT . In this way we prove the existence of a subsequence of Ψ ρn,h
= Ψn ρn,h that converges inL1loc([0, T]×R;R) to a functionψinL∞([0, T];BV(R; [−f(¯ρ), f(¯ρ)])) which satisfies
kψ(t+δ,·)−ψ(t,·)kL1(R;R)≤δ Lip(Ψ)CT . (29) For simplicity we still denote the subsequence Ψ ρn,h
. Since Ψ is invertible and Ψ−1 is continuous, alsoρn,hconverges in L1loc([0, T]×R;R) to a functionρ= Ψ−1(ψ) inL∞([0, T]×R; [0, R]). In particular, by (26) and (29) the estimates (11)
and (12) hold true.
Lemma 2. For any T >0
n,h→+∞lim Z T
0
Ξn,h(t)− Z
R−
w(x) ρ(t, x) dx
dt= 0 . Proof. LetT >0 and defineℓhT =⌊T /∆th⌋. Then by (20) and (25)
Z T 0
Ξn,h(t)− Z
R−
w(x) ρn,h(t, x) dx
dt≤
≤
ℓhT−1
X
ℓ=0
Z (ℓ+1)∆th
ℓ∆th
Ξn,hℓ (t−ℓ∆th)− Z
R−
w(x) ρn,hℓ+1(t−ℓ∆th, x) dx
dt +
Z T
ℓhT∆th
Ξn,h
ℓhT (t−ℓhT∆th)− Z
R−
w(x) ρn,h
ℓhT+1(t−ℓhT∆th, x) dx
dt
=
ℓhT−1
X
ℓ=0
Z ∆th
0
Z
R−
w(x)h
ρn,hℓ+1(0, x)−ρn,hℓ+1(t, x)i dx
dt +
Z T−ℓhT∆th
0
Z
R−
w(x)h ρn,hℓh
T+1(0, x)−ρn,hℓh
T+1(t, x)i dx
dt
≤
ℓhT−1
X
ℓ=0
Z ∆th
0
t dt+
Z T−ℓhT∆th
0
tdt
f(¯ρ) w(0−)
=∆t2h ℓhT + (T −ℓhT∆th)2
2 f(¯ρ) w(0−) .
Therefore, sinceℓhT∆th converges to T as h goes to infinity and ρn,h converges to ρ inL1loc(R+×R;R) as nand h go to infinity, the proof is complete.
Since ρn,h converges to ρ in L1loc, Proposition 4 and Lemma 2 imply that [t7→ρ(t)] satisfies the conditions (4a)–(4c) and (5) of Definition 1 with respect
to the problem (1). Moreover, ρ satisfies the condition (4d) of Definition 1 by Lemma 3, and it satisfies estimate (11) and (12) by Proposition 5. Finally, ob- serve that ρ ∈C0 R+;L1loc(R; [0, R])
because of entropy inequalities (4a)–(4c), see Ref. [3]. As already observed Ψ (ρ)∈C0(R+;BV(R;R)) thusρ(t)∈ D for all t.
Uniqueness of the entropy weak solutions to the Cauchy problem (1) in the case p∈Lip([0, R];R),ρ0∈ D follows directly from uniqueness in the L∞–framework, see the first part of the proof of Theorem 1.
4. The constrained Riemann problem
In this section we study constrained Riemann problems of the form
∂tρ+∂xf(ρ) = 0 (t, x)∈R+×R (30a)
f(ρ(t,0±))≤p Z
R−
w(x) ρ(t, x) dx
t∈R+ (30b)
ρ(0, x) =
ρL if x <0
ρR if x≥0 x∈R (30c)
withρL, ρR∈[0, R]. Along with(F)and (W), we assume that:
(P2) p belongs to PC([0, R] ; ]0, f(¯ρ)]) and is a non–increasing map.
The assumption (P2) is introduced in place of (P1) to allow an explicit con- struction of solutions to (30). However, the regularity of p required by (P2) is not enough to apply the results of Theorem 1. In fact, the uniqueness of entropy weak solutions as well as the stability estimate (10) do not hold in the present framework, as we will see in Example 2.
Aiming for a general construction of the solutions to (30), we allow p to be a multi–valued piecewise constant function, namely, see Fig. 1, right:
• there exist ξ1, . . . , ξn∈]0, R[ and p0, . . . , pn∈]0, f(¯ρ)], with ξi < ξi+1 and pi> pi+1, such thatp(0) =p0,p(R) =pn,p χ]ξi, ξi+1[ =pifori= 0, . . . , n, p(ξi) = [pi, pi−1] for i= 1, . . . , n, beingξ0 = 0 and ξn+1 =R.
Letσ(ρL, ρR) = (f(ρL)−f(ρR))/(ρL−ρR) be the speed of propagation of a shock between ρL and ρR, while λ(ρ) =f′(ρ) is the characteristic speed. Introduce the maps ˇρ,ρˆ : [0, f(¯ρ)]→[0, R] implicitly defined by
f(ˇρ(p)) =p=f(ˆρ(p)) and ρ(p)ˇ ≤ρ¯≤ρ(p)ˆ .
Let ˇρi= ˇρ(pi) and ˆρi= ˆρ(pi). Denote byRthe classical Riemann solver Refs. [23], [25]. This means that the map [(t, x)7→ R[ρL, ρR](x/t)] is the unique entropy weak solution for the unconstrained problem (30a), (30c), see Refs. [2], [29], [30] for its construction. As we will see in Proposition 6, the classical solutions given by R may not satisfy the constraint (30b). For this reason we consider also nonclassical solutions, namely solutions that do not satisfy the Lax entropy inequalities, see Ref. [24] as a general reference. In general, entropy weak solutions to (30) are not self–similar nor unique, as we will show in the two following examples.
Example 1. Let 0< ρL< ρR< Rbe such that f(ρL)> f(ρR). Ifξi ≤ρL< ξi+1, pi+1< f(ρR)< pi and j > iis such that f(ˆρj) =p(ˆρj) and f(ˆρk)> p(ˆρk) for all k∈ {i, . . . , j−1}, see Fig. 3, left, then the entropy weak solution to the correspond- ing Riemann problem (30)is not self–similar, see Fig. 3, right. More in detail, for sufficiently small times, the solution corresponds to the classical one and is given by a shock with speed σ(ρL, ρR)<0. The corresponding map [t7→ξ(t)] is increasing and by hypothesis, there exists a time t1 <−iw/σ(ρL, ρR) such that ξ(t1) =ξi+1. Then, the efficiency of the exit falls to pi+1 and the solution given by the classical Riemann solverRno longer satisfies the constraint condition (30b). As a result, at timet1 the solution performs a nonclassical discontinuity at the constraint location and two further classical shocks appear, one with speed σ(ρR,ρˆi+1) < 0 and one with speed σ(ˇρi+1, ρR) > 0. The final solution can then be constructed by taking into account the interactions between the shocks on each side of the constraint and the appearance of new shocks each time[t7→ξ(t)]crosses ξk,k∈ {i+ 1, . . . , j−1}.
Figure 3. Construction of a non self–similar entropy weak solution as in Example 1. On the left, the thick line corresponds to the efficiency of the exitp|]ρL,ˆρj[.
As we have seen, the lack of self–similarity is related to the jumps of [t 7→
p(ξ(t))]. Nevertheless, in the proof of Proposition 6 we show that any entropy weak solution of (30) is self–similar for sufficiently small times. Therefore, it makes sense to introduce nonclassicallocal Riemann solvers, see Definition 2. Then, the availability of a local Riemann solver allows us to construct a global solution to the Riemann problem (30) by a wave–front tracking algorithm in which the jumps in the map [t7→p(ξ(t))] are interpreted as interactions.
The next example shows that the entropy weak solutions to the constrained Riemann problem (30) are not necessarily unique.
Example 2. Consider the constrained Riemann problem (30) with ρL = ξi+1 ∈ ]¯ρ, R[ and ρR = ¯ρ. Assume that f(ˆρi+1) =pi+1 ≤ f(ξi+1) ≤pi =f(ˆρi) < f(¯ρ),
Figure 4. With reference to Example 2, the flux configuration and three different solutions ρ1, ρ2 and ρ3 to the same Riemann problem are represented from left to right. Here ˇρL= ˇρ(f(ρL)).
see Fig. 4, left, then
ρ1(x/t) =
R[ξi+1,ρˆi+1](x/t) if x <0 R[ˇρi+1,ρ](x/t)¯ if x≥0 , ρ2(x/t) =
ξi+1 if x <0
R[ˇρ(f(ξi+1)),ρ](x/t)¯ if x≥0 , ρ3(x/t) =
R[ξi+1,ρˆi](x/t) if x <0 R[ˇρi,ρ](x/t)¯ if x≥0 ,
are self–similar entropy weak solutions of problem (30) with the same datum, see Fig. 4. Clearly, the above solutions are distinct if pi+1 6= f(ξi+1) 6=pi, otherwise two of them may coincide. Additionally, for an arbitrarily chosen ¯t > 0, the functions
ρt,1¯ (t, x) =
ρ2(x/t) if0< t≤¯t ρ1(x/(t−t))¯ ift >¯t and x <0 ˇ
ρi+1 ift < t¯ ≤˜t1 and0≤x < σ(ˇρi+1,ρˇ(f(ξi+1))) (t−¯t)
¯
ρ ift < t¯ ≤˜t1 andx ≥σ(ˇρ(f(ξi+1)),ρ)¯ t ˇ
ρi+1 ift >˜t1 and 0≤x < σ(ˇρi+1,ρ) (t¯ −˜t1) + ˜x1
¯
ρ ift >˜t1 and x≥σ(ˇρi+1,ρ) (t¯ −˜t1) + ˜x1 ˇ
ρ(f(ξi+1)) otherwise,
ρt,3¯ (t, x) =
ρ2(x/t) if0< t≤¯t ρ3(x/(t−t))¯ ift >¯t and x <0 ˇ
ρi ift < t¯ ≤˜t3 and0≤x < σ(ˇρi,ρˇ(f(ξi+1))) (t−¯t)
¯
ρ ift < t¯ ≤˜t3 andx ≥σ(ˇρ(f(ξi+1)),ρ)¯ t ˇ
ρi ift >˜t3 and 0≤x < σ(ˇρi,ρ) (t¯ −˜t3) + ˜x3
¯
ρ ift >˜t3 and x≥σ(ˇρi,ρ) (t¯ −˜t3) + ˜x3 ˇ
ρ(f(ξi+1)) otherwise,
